Three Dimensional Manifolds All of Whose Geodesics Are Closed

John Olsen

A Dissertation

in

Mathematics

Presented to the Faculties of the University of Pennsylvania in Partial

Ful�llment of the Requirements for the Degree of Doctor of Philosophy

2009

Wolfgang Ziller

Supervisor of Dissertation

Tony Pantev

Graduate Group Chairperson

Acknowledgments

The �rst part of this work was supervised by Burkhard Wilking in Münster. I would

like to thank Burkhard for proposing the problem to me and for helpful advice

on how to attack it. The second part of my work was supervised by Wolfgang

Ziller in Philadelphia. Thank you, Wolfgang, for numerous helpful and enlightening

conversations, and much helpful advice on how to proceed with my work. Most

of all I would like to thank you for creating an inviting and pleasant atmosphere

in which it was a pleasure to work and where I felt welcome whenever I came to

see you. The idea of getting all your students and visitors together for a seminar

is excellent! I really enjoyed these �secret seminars� and learned a lot both from

giving talks and from listening to other people's talks. A big thank you to former

and current participants, Corey, Kris, Martin, Chenxu, Ricardo, Jason and Will. A

particular thanks to Martin for being my tutor for the time when I was working with

biquotients!

Part of this work was done while visiting Karsten Grove at the University of

Maryland. Thank you, Karsten, for many great conversations and for conveying

ii

your love of doing mathematics to me at a time when I needed exactly that! Our

Monday meetings were always something I looked forward to, and, of course, I also

want to thank both Fernando and Joe for making them so enjoyable.

It was during my graduate studies at the University of Aarhus I became interested

in topology and geometry and I would like to thank Ib Madsen, Marcel Bökstedt,

Jørgen Tornehave and Johan Dupont for sparking that interest through great courses

and numerous conversations. I would especially like to thank Marcel Bökstedt for

the many great conversations and discussions we have had over the years.

I would like to thank our great secretaries Janet, Monica, Paula and Robin.

Everything around here runs smoothly and you are the reason for that! Janet put

in a huge e�ort to help us get our job applications ready and helping us out with

day-to-day graduate a�airs. I am very thankful for all her help! I don't even have

to say anything to make Monica laugh. What a great skill to have! I want to thank

Monica for enduring my endless number of more or less ridiculous questions and

answering them swiftly and accurately. Figuring out how the system works is a

whole lot easier with her around! Thank you to Paula and Robin for all their help!

During my �rst year at the University of Pennsylvania, I shared an o�ce with

Vittorio. We will never agree 100% on who is the best bike rider or what the nicest

Italian city might be, but that only added to the conversations! Thanks to Vittorio

for a fun year! The next two years I shared an o�ce with Andrew. I couldn't have

asked for a better o�ce mate! Thanks to Andrew for two great years!

iii

I want to thank Elena, Enka and Sarah for introducing me to the �ne art of tea

drinking, and for numerous fun hours chatting in Enka's o�ce. I also want to thank

Lee, Sohrab and Torcuato for many helpful and enjoyable nights in the �rst year

o�ce. Thanks to Tobi for great company on the bike climbing the Belmont Hills.

Those were fun times! Thanks to Clay, Jen and Shea for being such good colleagues!

I also want to thank my friends Clarisse and Tom for their wonderful company at

our weekly dinners!

Finally, I want to thank my Mom Lilli for her great support from 6000km away.

I certainly would not have �nished any kind of advanced degree if you had not been

so focused on getting us the best education and making sure we worked hard in

school. I am both grateful and impressed that you managed that! Thank you also

to my brother Erik for setting the bar high, both academically and on the road, and

for your interest in my work.

iv

ABSTRACT

Three Dimensional Manifolds All of Whose Geodesics Are Closed

John Olsen

Wolfgang Ziller, Advisor

We present some results concerning the Morse Theory of the energy function on

the free loop space of S3 for metrics all of whose geodesics are closed. We also show

how these results may be regarded as partial results on the Berger Conjecture in

dimension three.

v

Contents

1 Introduction 1

2 Preliminaries 6

3 Rational S1-equivariant Perfectness of the Energy Function 11

4 Orientability of Negative Bundles 25

5 Topology of the Three Dimensional Critical Manifolds 27

6 Index Growth 31

7 Reductions of the Berger Conjecture in Dimension Three 34

vi

Chapter 1

Introduction

About 30 years ago Berger conjectured that on a simply connected manifold all of

whose geodesics are closed, all geodesics have the same least period. In addition

to the spheres and projective spaces with the standard metrics, the so-called Zoll

metrics on Sn have this property as well; see [Bes78, Corollary 4.16]. The weaker

statement that there exists a common period is a special case of a theorem due to

Wadsley; see [Bes78, Theorem 7.12]. The lens spaces with the canonical metrics

show that simply connectedness is necessary. On S2n+1/Zk, k > 2 all geodesics are

closed with common period 2π, but there exist geodesics of smaller period.

Bott and Samelson studied the topology of such manifolds and showed that they

must have the same cohomology ring as a compact rank one symmetric space. In

1982 Gromoll and Grove proved the Berger Conjecture for metrics on S2, [GG82,

Theorem 1].

1

In this thesis we present some results on the Morse theory on the free loop space

of S3 for metrics all of whose geodesics are closed. We also see how these results

may be regarded as partial results on the Berger Conjecture.

Before we state the results we will review some basic notions from Morse theory

on the free loop space of a Riemannian manifold; see Chapter 2 for an introduction.

Let M be a Riemannian manifold and let the free loop space ΛM be the set of

absolutely continuous maps c : S1 → M with square integrable derivative and let

the energy function E: ΛM → R be given by E(c) =∫ 2π

0|ċ(t)|2dt. The free loop

space ΛM can be given the structure of a smooth Hilbert manifold which makes E

into a smooth function. It follows from the �rst variation formula that the critical

points of E are the closed geodesics on M . The group O(2) acts on the free loop

space by reparameterization and since the action leaves E invariant, a critical point

is never isolated. If the critical sets are submanifolds of ΛM , then we say that such

a manifold N is nondegenerate if the null space of the Hessian Hessc(E) is equal to

the tangent space TcN . If all critical manifolds are nondegenerate in this sense we

say that E is a Morse-Bott function.

Let ΛaM be the set E−1([0, a]) ⊆ ΛM . If N is the only critical submanifold of

energy a, one can use the gradient �ow to show that there is a homotopy equiva-

lence Λa+�M ' Λa−�M ∪f D(ξ−) for some gluing map f : S(ξ−) → Λa−�M , where

ξ− is the negative bundle over N whose �ber consists of the sum of the negative

eigenspaces of Hessc(E). The rank of ξ− is denoted λ(N) and is called the in-

2

dex of the critical manifold. The spaces D(ξ−) and S(ξ−) are the disk respec-

tively sphere bundle of ξ−. Excision gives an isomorphism H i(Λa+�M, Λa−�M ; R) ∼=

H i(D(ξ−), S(ξ−); R) and if the negative bundle over N is orientable the Thom iso-

morphism yields H i(D(ξ−), S(ξ−); R) ∼= H i−λ(N)(N ; R) for any coe�cient ring R. If

N is not orientable the last isomorphism holds with Z2 coe�cients. If R is a �eld

we say that E is perfect if Hj(ΛM ; R) =⊕

j Hj−λ(Nj)(Nj; R), where the sum is over

all critical manifolds.

For a topological group G we let EG be a contractible topological space on

which G acts freely and let EG/G = BG be the classifying space of G. For a G-

space X the G-equivariant cohomology of X is de�ned to be the usual cohomology

of the quotient space (X × EG)/G = XG. While the action of G on X might not

be free, the diagonal action of G on X × EG is always free, and the equivariant

cohomology of X models the cohomology of X/G is the sense that for a free action

we have H∗G(X; R)∼= H∗(X/G; R). The negative bundles are O(2)-vector bundles

and one can for any group G ⊆ O(2) do equivariant Morse Theory analogously to the

ordinary theory. In particular, one gets the isomorphism H iG(Λa+�M, Λa−�M ; R) ∼=

H iG(D(ξ−), S(ξ−); R), and if the G-equivariant negative bundle is orientable, the

Thom isomorphism yields H iG(D(ξ−), S(ξ−) ∼= H i−λ(N)G (N ; R) for any ring R. If R

is a �eld, we say that E is perfect with respect to G-equivariant cohomology with

coe�cients in R if HjG(ΛM ; R) =⊕

j Hj−λ(Nj)G (Nj; R), where the sum is over all

critical manifolds.

3

We can now state the main results of the thesis.

Theorem 1. Let g be a metric on S3 all of whose geodesics are closed. Then

relative to the point curves the energy function E is perfect with respect to rational

S1-equivariant cohomology.

The fact that the energy function is a Morse-Bott function was observed by

Wiking; see [Wil01, Proof of Step 3]. This allows us to do Morse Theory on ΛS3.

The next theorem allows us to use arbitrary coe�cients for the homology.

Theorem 2. Let g be a metric on S3 all of whose geodesics are closed. Then the

negative bundles over the critical manifolds are orientable both as ordinary and S1-

equivariant vector bundles.

We have the following structure result for the critical manifolds of the energy

function.

Theorem 3. Let g be a metric on S3 all of whose geodesics are closed. A critical

manifold in ΛS3 is either di�eomorphic to the unit tangent bundle of S3 or has the

integral cohomology ring of a three dimensional lens space S3/Z2k.

The three theorems together imply the following theorem; see Chapter 7, Theo-

rem 3.

Theorem 4. Let g be a metric on S3 all of whose geodesics are closed. The geodesics

have the same least period if and only if the energy function is perfect for ordinary

cohomology.

4

One possible way to attack the Berger Conjecture in dimension three is to assume

that there exist exceptional closed geodesics of period 2π/n for n > 1 and then try

to derive a contradiction by using the results above together with the Bott Iteration

Formula, which calculate the index of an iterated geodesic.

We show that there exists only one critical manifold N of index two and that this

critical manifold must be three dimensional. By the Bott Iteration Formula we see

that for c ∈ N , the index of c2 is either four or six. If we assume that the sectional

curvature K satis�es 1/4 ≤ K ≤ 1 we are able to get a contradiction in the case

where Index(c2) = 4 by using a result from [BTZ83]. We have not yet been able to

handle the case where Index(c2) = 6; see Chapter 7 Theorem 4 for a more detailed

description of further properties we are able to prove.

5

Chapter 2

Preliminaries

We will review some basic notions from Morse theory on the free loop space of a

Riemannian manifold. The standard reference is [Kli78]; see also [BO04, Chapter 3]

and [Hin84] for a short introduction.

Let M be a Riemannian manifold and let ΛM = W 1,2(S1, M) be the free loop

space of M . The free loop space ΛM can be given the structure of a smooth Hilbert

manifold which makes E into a smooth function; see [Kli78, Theorem 1.2.9]. Note

that there is a natural inclusion W 1,2(S1, M) → C0(S1, M) and a fundamental

theorem states that this map is a homotopy equivalence; see [Kli78, Theorem 1.2.10].

The tangent space at a point c ∈ ΛM , TcΛM , consists of all vector �elds along c of

class W 1,2, and is a real Hilbert space with inner product given by

〈〈X, Y 〉〉1 =∫ 2π

0

〈X(t), Y (t)〉+ 〈DXdt

,DY

dt〉dt,

where Ddt

is the covariant derivative along c induced by the Levi-Civita connection

6

on M . Let the energy function E: ΛM → R be given by E(c) =∫ 2π

0|ċ(t)|2dt. It

follows from the �rst variation formula that the critical points of E are the closed

geodesics on M .

Let N be a critical manifold of E in ΛM and let c ∈ N be a critical point. We

use the convention that the critical sets are maximal and connected throughout. By

de�nition N satis�es the Bott nondegeneracy condition if TcN = Ker(Hessc(E)). If

all critical manifolds are nondegenerate in this sense, the energy function is called

a Morse-Bott function. What this says geometrically is that the dimension of the

space of periodic Jacobi vector �elds along the geodesic c is equal to the dimension

of the critical manifold.

Now assume that Nj, j = 1, . . . , l, are the critical manifolds of energy a, that

they satisfy the Bott nondegeneracy condition, and that there are no other critical

values in the interval [a − �, a + �], � > 0. The metric on TcΛM induces a splitting

of the normal bundle, ξ, of Nj in ΛM into a positive and a negative bundle, ξ =

ξ+ ⊕ ξ−, such that the Hessian of the energy function is positive de�nite on ξ+ and

negative de�nite on ξ−. Furthermore λ(Nj) = rank ξ− is called the index of Nj

and is �nite. Note that the index is constant on each critical manifold, since it is

connected. We denote by ΛaM the set E−1([0, a]) ⊆ ΛM . Let D(ξ−(Nj)) = D(Nj)

and S(ξ−(Nj)) = S(Nj) be the disc, respectively sphere, bundle of the negative

bundle ξ− over Nj. Using the gradient �ow, one shows that there exists a homotopy

equivalence Λa+�M ' Λa−�M ∪f ∪lj=1D(Nj) for some gluing maps fj : S(Nj) →

7

Λa−�M ; see [Kli78, Theorem 2.4.10].

If the negative bundle over Nj is orientable for all j, excision and the Thom

isomorphism yield

H i(Λa+�M, Λa−�M ; R) ∼=l⊕

j=1

H i(D(Nj), S(Nj); R) ∼=l⊕

j=1

H i−λ(Nj)(Nj; R)

for any coe�cient ring R. If N is not orientable the isomorphism holds with Z2

coe�cients. The cohomology of Λa+�M is now determined by the cohomology of

Λa−�M and Nj by the long exact cohomology sequence for the pair (Λa+�M, Λa−�M).

This way one can in principle inductively calculate the cohomology of ΛM from the

cohomology of the critical manifolds. If the map Λa+�M → (Λa+�M, Λa−�M) induces

an injective map H i(Λa+�M, Λa−�M ; R) → H i(Λa+�M ; R) for all i, we say that all

relative classes can be completed to absolute classes. This is equivalent to all the

boundary maps in the long exact cohomology sequence for the pair (Λa+�M, Λa−�M)

being zero. If this holds for all i and all critical values a, we say that E is perfect. If

R is a �eld, perfectness implies that Hj(ΛM ; R) =⊕

j Hj−λ(Nj)(Nj; R), where the

sum is over all critical manifolds.

For a topological group G we let EG be a contractible topological space on which

G acts freely and let EG/G = BG be the classifying space of G. For a G-space X

the quotient space (X ×EG)/G = X ×G EG = XG is called the Borel construction.

The G-equivariant cohomology of X is de�ned to be the usual cohomology of the

Borel construction XG. We have a �bration X → X ×G EG → BG and, if the

G-action on X is free, a �bration X×G EG → X/G with �ber EG, i.e. the map is a

8

(weak) homotopy equivalence. If G acts on a manifold X with �nite isotropy groups,

the map X ×G EG → X/G is a rational homotopy equivalence and the cohomology

of X/G is concentrated in �nitely many degrees when X is �nite dimensional and

compact.

For G a group acting on ΛM we consider G-equivariant Morse theory. Let

ξ → X be a vector bundle where G acts on ξ such that p : ξ → X is equivariant

and the action is linear on the �bers. For such vector bundles we de�ne the G-

vector bundle by ξG = ξ ×G EG → XG. The G-vector bundle ξG is orientable if

and only if ξ is orientable and G acts orientation preserving on the �bers. Thus

if G is connected ξG is orientable if and only if ξ is orientable. For an oriented

rank k G-vector bundle ξG over X there is a G-equivariant Thom isomorphism,

H∗G(D((X)), S((X)); R)∼= H∗−kG (X; R).

The action of O(2) on S1 induces an action of O(2) on the free loop space via

reparametrization. Since the energy function is invariant under the action of O(2)

and O(2) acts by isometries with respect to the inner product used to de�ne the

negative bundles, the negative bundles are O(2)-bundles in this sense. For any

group G ⊆ O(2) we have similarly to the above

H iG(Λa+�M, Λa−�M ; R) ∼=

l⊕j=1

H iG(D(Nj), S(Nj); R)∼=

l⊕j=1

Hi−λ(Nj)G (Nj; R)

where as above D(Nj) and S(Nj) are the disk and sphere bundle of the nega-

tive bundles ξ− over the critical manifold Nj j = 1, . . . , l and where a is the

energy of the critical manifolds Nj. The second isomorphism holds for any co-

9

e�cient ring R if the negative bundle is oriented and with Z2 coe�cients if it

is nonorientable. The G-equivariant cohomology of Λa+�M is determined by the

G-equivariant cohomology of Λa−�M and Nj by the long exact G-equivariant co-

homology sequence for the pair (Λa+�M, Λa−�M). In principle this allows us to

inductively calculate the G-equivariant cohomology of ΛM from the G-equivariant

cohomology of the critical manifolds. Again we say that E is perfect with respect to

G-equivariant cohomology if the map Λa+�M → (Λa+�M, Λa−�M) induces an injec-

tive map H iG(Λa+�M, Λa−�M ; R) → H iG(Λa+�M ; R) for all i and for all critical values

a. If R is a �eld, perfectness implies that HjG(ΛM ; R) =⊕

j Hj−λ(Nj)G (Nj; R), where

the sum is over all critical manifolds. See [Hin84] for an introduction to equivariant

Morse theory on ΛM .

10

Chapter 3

Rational S1-equivariant Perfectness

of the Energy Function

In the case of the canonical metric g0 on S3 we let Bk ∼= T 1S3 be the manifold of

k-times iterated geodesics. The energy function is a Morse-Bott function for the

metric g0. The manifolds Bk, k ∈ N>0 and the point curves S3 are the only critical

manifolds for E and the induced action of S1/Zk on Bk is free. The S1-equivariant

cohomology of T 1S3 can thus be calculated from the Gysin sequence for the bundle

S1 → T 1S3 → T 1S3/S1 and is given by

H iS1(T1S3;Q) =

Q, i = 0, 4,

Q2, i = 2,

0, otherwise.

11

Since by [Zil77] the indices of the critical manifolds Bk in ΛS3 are 2(2k−1), k ∈ N>0,

we see that for all k the rational S1-equivariant cohomology of Bk is concentrated

in even degrees. Hence by the Lacunary Principle the energy function is perfect

relative to S3 for rational S1-equivariant cohomology.

Proposition 3.1. The rational S1-equivariant cohomology of ΛS3 relative to S3 is

given by

H iS1(ΛS3, S3;Q) =

Q, i = 2

Q2, i = 2k, k ∈ N, k > 1

0, otherwise.

More generally, Hingston calculated the S1-equivariant cohomology of the free

loop space of any compact rank one symmetric space; see [Hin84, Section 4.2].

Let g be a metric on S3 all of whose geodesics are closed and normalized so that

2π is the least common period. We assume that there exist exceptional geodesics

on (S3, g) of period 2π/n for n > 1. Since all geodesics are closed with common

period 2π the geodesic �ow de�nes an e�ective and orientation preserving action of

R/2πZ = S1 on T 1S3. Choose a metric on T 1S3 such that the action of S1 becomes

isometric. The full unit tangent bundle corresponds to geodesics of period 2π and

the closed geodesics of length (k/n)2π can be identi�ed with the �xed point set of

the element e2πi/n ∈ S1 in T 1S3, i.e. the �xed point set of Zn ⊆ S1. The �xed

point set of Zn ⊆ S1 again has an e�ective and orientation preserving action of

S1/Zn = S1 since S1 is abelian. Since the critical sets of the energy function can be

12

identi�ed with the �xed point sets of some g ∈ S1 and the metric is chosen so that

the action is isometric, it follows that the critical sets are compact, totally geodesic

submanifolds of T 1S3.

We recall an observation made by Wilking, [Wil01, Proof of Step 3]: If all

geodesics are closed, the energy function is a Morse-Bott function. To see this, one

notes that a critical manifold N ⊆ T 1S3 of geodesics of length 2π/n is a connected

component of the �xed point set of an element g = e2πi/n ∈ S1. The dimension of

the critical manifold is equal to the multiplicity of the eigenvalue 1 of the map g∗v

at a �xed point v. Since the di�erential of the geodesic �ow is the Poincaré map,

the multiplicity of the eigenvalue 1 at v is equal to the dimension of the vector space

of 2π/n-periodic Jacobi �elds along the geodesic c(t) = exp(tv), t ∈ [0, 2π/n] (ċ

is considered as a periodic Jacobi �eld as well). Since the null space of Hessc(E)

consists of 2π/n-periodic Jacobi vector �elds, we see that the kernel of Hessc(E) is

equal to the tangent space of the critical manifolds.

We know that the geodesic �ow acts orientation preserving on the �xed point

sets, but we need to show that the �xed point sets are indeed orientable. This and

the fact that the �xed point sets have even codimension follow from the following

lemma.

Lemma 3.1. The �xed point sets Fix(Zn) ⊆ T 1S3 have even codimension and are

orientable.

Proof. We recall a few facts about the Poincaré map; see [BTZ82, page 216]. Let c

13

be a closed geodesic with c′(0) = v 6= 0. Let ϕt denote the geodesic �ow and note

that a closed geodesic with v = c′(0) corresponds to a periodic orbit ϕtv. The �ow

ϕt maps the set TrS3 = {v ∈ TS3 | |v| = r} into itself. The Poincaré map P of c is

the return map of a local hypersurface N ⊂ TrS3 transversal to v. The linearized

Poincaré map is up to conjugacy independent of N and is given by P = DvP. One

can choose N such that TvN = V ⊕ V , where V = v⊥ ⊆ TpS3, where p is the

footpoint of v. On V ⊕ V there is a natural symplectic structure which is preserved

by P . As above we identify the critical manifolds with the �xed point sets of Zn ⊂ S1

and notice that since the critical manifolds are nondegenerate, the dimension of the

�xed point sets is equal to the multiplicity of the eigenvalue 1 of P plus one. By

[Kli78, Proposition 3.2.1] we know that the multiplicity of 1 as an eigenvalue of P is

even. Hence the dimension of Fix(Zn) ⊆ T 1S3 is odd and the codimension is even.

If the �xed point sets are one or �ve dimensional, they are clearly orientable, so

assume dim Fix(Zn) = 3. In this case the normal form of P has a 2 × 2 identity

block and a 2 × 2 block which is a rotation, possibly by an angle π. Denote the

two dimensional subspaces by Vid and Vrot. Using the symplectic normal form for P

we know that Vid and Vrot are orthogonal with respect to the symplectic form and

that the restriction of the symplectic form to each subspace is nondegenerate; see

[BTZ82, page 222]. If we consider the normal bundle ν Fix(Zn) with �ber νv Fix(Zn)

for v ∈ Fix(Zn), we have νv Fix(Zn) = Vrot. Since Vrot is a symplectic subspace, it

carries a canonical orientation. This gives a canonical orientation on each �ber of

14

the normal bundle, which shows that the normal bundle is orientable and hence that

Fix(Zn) is orientable.

The one dimensional critical manifolds are di�eomorphic to circles. If c is a one

dimensional critical manifold, the geodesic c(t) = c(−t) is a second critical manifold

of the same index. If the critical manifold is �ve dimensional it is the full unit tangent

bundle. If the exceptional critical manifold that consists of prime closed geodesics

is �ve dimensional then all geodesics are closed with period 2π/n contradicting the

assumption.

We now state and prove the main result of the thesis.

Theorem 1. Let g be a metric on S3 all of whose geodesics are closed. Then

relative to the point curves the energy function E is perfect with respect to rational

S1-equivariant cohomology.

Proof. The proof takes up several pages, and we �rst give a short outline. We will

use the Index Parity Theorem repeatedly; see [Wil01, Theorem 3]. The Index Parity

Theorem states that for an oriented Riemannian manifold Mn all of whose geodesics

are closed, the index of a geodesic in the free loop space is even if n is odd, and odd

if n is even; in particular, in our case it states that all indices are even. The idea is

to show that the contributions H iS1(D(N), S(N);Q) from a critical manifold N to

the S1-equivariant cohomology of ΛS3 occur in even degrees only. If the negative

bundle over N is orientable this is by the Thom Isomorphism equivalent to showing

15

that H iS1(N ;Q) = 0 for i odd. Perfectness of the energy function then follows from

the Lacunary Principle, since the critical manifolds all have even index.

We �rst show that the negative bundles over the one and �ve dimensional crit-

ical manifolds are oriented and that the critical manifolds only have S1-equivariant

cohomology in even degrees. If the critical manifold is three dimensional we �rst

use Smith Theory to show that the S1-Borel construction is rationally homotopy

equivalent to S2. If the negative bundle is oriented the contributions occur in even

degrees only. If the negative bundle is nonorientable we use Morse Theory and a

covering space argument to show that the critical manifold does not contribute to

the S1-equivariant cohomology.

We begin the proof by considering the �ve dimensional case. The �ve dimensional

critical manifold is di�eomorphic to the unit tangent bundle. It is clear that the

negative bundles over the unit tangent bundle are oriented, since the unit tangent

bundle is simply connected. The Borel construction T 1S3 ×S1 ES1 is rationally

homotopy equivalent to T 1S3/S1, since the action has �nite isotropy groups, and

thus the possible degrees where T 1S3 ×S1 ES1 has nonzero rational cohomology

is zero through four. Using these facts and the Gysin sequence for the bundle

S1 → T 1S3 × ES1 → T 1S3 ×S1 ES1 we see that H∗S1(T 1S3;Q) = H∗(S2 × S2;Q)

and hence has nonzero classes in even degrees only.

Next, we show that the negative bundles over the one dimensional critical man-

ifolds are oriented and thus, since the S1-Borel construction of the one dimensional

16

critical manifolds has rational cohomology as a point, the contributions occur in

even degrees only.

Proposition 3.2. The negative bundles over the one dimensional critical manifolds

are (S1-equivariantly) orientable.

Proof. The negative bundle over the critical manifold consisting of a prime closed

geodesics c is orientable, since we can de�ne an orientation in one �ber of the negative

bundle and use the free S1/Zn-action to de�ne an orientation in the other �bers.

Let ξn be the negative bundle over the n-times iterated geodesic. The repre-

sentation of Zn on the �bers of ξn is presented in [Kli78, Proposition 4.1.5]. The

representation of Zn is the identity on a subspace of dimension Index(c) (which cor-

responds to the image of the bundle ξ1 under the n-times iteration map) and is a

sum of two dimensional real representations given by multiplication by e±2πip/n on

a vector space of even dimension. If n is even, Zn also acts as − id on a subspace of

dimension Index(c2)− Index(c).

We know by [Kli78, Proposition 4.1.5] that the dimension of the subspace on

which a generator T of Zn acts as − id is equal to Index(c2)− Index(c), which by the

Index Parity Theorem is even. By [Kli78, Lemma 4.1.4] the pair (Dk/Zn, Sk−1/Zn)

is orientable (k = Index(cn)), since the dimension of the subspace on which T acts

as − id is even dimensional. Pick an orientation in one �ber of the negative bundle

invariant under the action of Zn. Use the S1/Zn- action to de�ne an orientation in

any other �ber of the negative bundle. Since the orientation is chosen to be invariant

17

under the action of Zn the orientation on the negative bundle is well-de�ned.

We now treat the three dimensional case and �rst prove the following important

fact.

Proposition 3.3. Assume that the critical manifold N has dimension three. Then

the S1-Borel construction of N is rationally homotopy equivalent to S2.

Proof. Assume that Zn acts trivially on N . We start by considering the quotient

N → N/(S1/Zn). The action of S1/Zn is e�ective and has isotropy at a closed

geodesic which is an lth iterate, for some l, of a closed geodesic in a one dimensional

critical manifold. The S1/Zn-action normal to the S1/Zn-orbit acts as a rotation

by e2πi/l. Hence N/(S1/Zn) is a two dimensional orbifold, since a neighborhood of

an arbitrary point in N/(S1/Zn) is homeomorphic to R2/Zl. However, the quotient

R2/Zl is homeomorphic to R2 so in particular N/(S1/Zn) is homeomorphic to a

surface. Since by Lemma 3.1 Fix(Zn) is orientable and the action of S1/Zn is

orientation preserving, the quotient is an orientable surface of genus g.

Using the Gysin sequence we can calculate H1(N ;Q) from the bundle S1 →

N × ES1 → N ×S1 ES1, since we know that N/(S1/Zn) is rationally homotopy

equivalent to N ×S1 ES1. The Gysin sequence yields

H i(N ;Q) =

Q i = 0, 3

Q2g, i = 1, 2 if χ 6= 0,

Q2g+1, i = 1, 2 if χ = 0,

18

where χ denotes the Euler class of the bundle.

If n is a prime, p, we can use Smith theory for the Zn-action on T1S3 to bound

the sum of the Betti numbers of the �xed point sets, i.e. we know that the sum∑bi(Fix(Zp);Zp) ≤

∑bi(T

1S3;Zp) = 4 for an arbitrary prime p; see [Bre72, Theo-

rem 4.1]. Hence we have∑

bi(Fix(Zp);Q) ≤ 4, by the Universal Coe�cient Theo-

rem. If n is not prime we choose a p such that p|n and such that Fix(Zp) is three

dimensional, which is possible since N is three dimensional and the only critical

manifold of dimension �ve is T 1S3. Then N ⊆ Fix(Zp) and since both N and

Fix(Zp) are closed three dimensional submanifolds of T1S3, N equals a component

of Fix(Zp). Hence we have∑

bi(N ;Q) ≤ 4, which implies that g = 0.

Thus, if the negative bundle is orientable we only have contributions in even

degrees. We now treat the case where the negative bundle is nonorientable.

Proposition 3.4. Let ξ− → N be a nonorientable negative bundle over a three

dimensional critical manifold N . Then the cohomology groups H∗S1(D(N), S(N);Q)

vanish.

Proof. Let p : Ñ → N be the twofold cover such that the pull-back p∗ξ− = ξ̃− is

orientable. Let ι be the covering involution on Ñ which lifts to ξ̃−. Note that the

S1-equivariant negative bundle ξ−S1 → NS1 is also nonorientable. Lift the action of

S1/Zn, or possibly a twofold cover of S1/Zn, on ξ

− to an action on ξ̃− such that

ι becomes equivariant with respect to this action. Let ξ̃−S1 → ÑS1 be the oriented

19

twofold cover of ξS1 → NS1 . We know from Proposition 3.3 that NS1 is rationally

homotopy equivalent to S2. The action of S1/Zn on Ñ also has �nite isotropy

groups, so by an argument similar to the one in the proof of Proposition 3.3 we see

that Ñ/(S1/Zn) is an oriented surface of genus g, Fg. The Borel construction ÑS1

is rationally homotopy equivalent to Fg.

There are two cases to consider: g = 0 and g > 0. We �rst treat the case g = 0.

Since the bundle ξ̃−S1 is orientable there exists a Thom class and this cohomology

class changes sign under the action of ι∗, since otherwise it would descend to give a

Thom class for ξ−S1 making it orientable. By general covering space theory we know

that H∗S1(D(Ñ), S(Ñ);Q)ι∗ = H∗S1(D(N), S(N);Q) and H

∗S1(Ñ ;Q)

ι∗ = H∗S1(N ;Q),

where the superscript ι∗ denotes the classes that are �xed under the action of ι∗,

and where as before, D(N) = D(ξ(N)) etc.

By the Thom isomorphism we have H iS1(D(Ñ), S(Ñ);Q) = Hi−λ(N)S1 (Ñ ;Q) and

since g = 0 we have H iS1(Ñ ;Q) = 0 for all i 6= 0, 2, which means that we only

have to see what happens to the degree zero and degree two cocycles. By the

above, this implies that H iS1(D(N), S(N);Q) = 0 for i 6= λ(N), λ(N) + 2. The

action of ι∗ on H0S1(Ñ ;Q) is trivial and since the Thom class changes sign, the

action of ι∗ on Hλ(N)

S1 (D(Ñ), S(Ñ);Q) has no �xed points. Hence we conclude that

Hλ(N)

S1 (D(N), S(N);Q) = 0.

Since both N and Ñ are two spheres we have H2S1(Ñ ;Q) = Q and H2S1(N ;Q) =

Q, but also that H2S1(Ñ);Q)ι∗ = H2S1(N ;Q) = Q. This implies that ι

∗ acts trivially

20

on H2S1(Ñ ;Q) and, since the Thom class changes sign under the action of ι∗, as − id

on Hλ(N)+2

S1 (D(Ñ), S(Ñ);Q). We conclude that Hλ(N)+2

S1 (D(N), S(N);Q) = 0.

The second case to consider is g > 0. In that case we will derive a contradiction.

Let 2k be the minimal index of a critical manifold on which Zn acts trivially and

whose negative bundle is nonorientable. If there is more than one critical manifold

of index 2k with nonorientable negative bundle, we repeat the argument for each

manifold. If the action of S1/Zn had been free, NS1 would be homotopy equivalent

to S2 and the negative bundle would have been orientable. Hence the action of

S1/Zn on N has �xed points and since the action is e�ective, the �xed points are

one dimensional critical manifolds.

Since the map Ñ/S1 → N/S1 is a branched covering, the Riemann-Hurwitz

formula implies that there are 2g + 2 branched points. Exceptional orbits of the

S1-action on Ñ are circles, since the action is e�ective, and they project down to

exceptional orbits (shorter geodesics) for the action on N . Branched points for the

covering Ñ/S1 → N/S1 correspond to orbits where the isotropy of the action on N

is bigger than for the action on Ñ . Hence we see that the branched points come

from exceptional orbits for the action on N and hence that there are at least 2g + 2

exceptional orbits in N .

The exceptional orbits are shorter geodesics, and since by the Bott Iteration

Formula Index(cq) ≥ Index(c), these circles must all have index less than or equal

to 2k. By the following lemma we deduce that the index must be equal to 2k. For

21

later reference we state this lemma separately.

Lemma 3.2. There exists no one dimensional critical manifolds of index less than

2k.

Proof. Assume for contradiction that there exists a one dimensional critical manifold

of index 2h < 2k, h ≥ 2. This critical manifold hence contributes a Q2 in degree

2h. A three dimensional critical manifold contributes a Q in the degree equal to

the index and a Q in degree equal to the index plus 2, since the negative bundle

is orientable (hence a total of two classes). The unit tangent bundle contributes

a total of 4 classes. There are no cancellations in degree less than 2k, since the

contributions all have even degrees. The total number of classes needed to "�ll the

gaps" (from H2S1(ΛS3, S3;Q) to H2h−2S1 (ΛS

3, S3;Q)) in the cohomology is 2h−3, since

H2S1(ΛS3, S3;Q) = Q. Notice that as 2h < 2k no cancellations are possible, since all

contributions occur in even degrees. This implies that no three dimensional critical

manifold can have index 2h−2 and the unit tangent bundle cannot have index 2h−2

or 2h−4, since otherwise dim H2hS1(ΛS3, S3;Q) ≥ 3, since we have contributions from

the two circles and the three or �ve dimensional critical manifold. Hence we have to

�ll an odd number of holes with contributions that only come in pairs or quartets.

This is clearly impossible.

If h = 1 we get a contribution of Q2 in degree 2, which cannot cancel out, since

the index two critical manifold only contributes in even degrees. This contradicts

the fact that H2S1(ΛS3, S3;Q) = Q.

22

By a similar argument, we see that there is at most one three or �ve dimensional

critical manifold for each even index less than 2k. If the unit tangent bundle has

index 2j, there is no critical manifold of index 2j + 2, since H2S1(T1S3;Q) = Q2.

Now we derive the contradiction. This is done by considering the possible con-

tributions to H2kS1(ΛS3, S3;Q). The three dimensional critical manifold of index

2k − 2 contributes a Q in degree 2k, a �ve dimensional critical manifold of in-

dex 2k − 4 contributes a Q in degree 2k, and similarly a �ve dimensional critical

manifold of index 2k − 2 contributes a Q2 in degree 2k. Hence we get a contribu-

tion to H2kS1(ΛS3, S3;Q) of at least a Q from the critical manifold of index 2k − 2

or 2k − 4 and a Q2g+2 from the exceptional orbits. By the Thom isomorphism

we have H2k+1S1 (D(Ñ), S(Ñ);Q) = H1S1(Ñ ;Q) and furthermore we have by cov-

ering space theory that H2k+1S1 (D(Ñ), S(Ñ);Q)ι∗ = H2k+1S1 (D(N), S(N);Q), which

implies that H1S1(Ñ ;Q)ι∗ = H2k+1S1 (D(N), S(N);Q). Since H

1S1(Ñ ;Q) = Q

2g, we

see that the maximal contribution to H2k+1S1 (ΛS3, S3;Q) is Q2g, which for example

happens if ι∗ acts as − id on Q2g. This yields a contradiction since we now have

dim H2kS1(ΛS3, S3;Q) ≥ 3.

If there exists another critical manifold of index 2k with nonorientable negative

bundle we repeat the argument above. The exceptional geodesics that contribute in

degree 2k are distinct from the ones in other critical manifolds since the iterates of

the shorter geodesics lie in di�erent connected critical manifolds.

This �nishes the proof that E is perfect.

23

A consequence of the perfectness of the energy function is the following general

fact.

Corollary 3.1. The minimal index of a critical manifold consisting of nonconstant

geodesics is two. For every number 2k there exists at most one connected critical

manifold of index 2k for every k ≥ 1.

Proof. As a global minimum of E, S3 has index zero. If one of the critical manifolds

N consisting of nonconstant geodesics has index zero, H0S1(ΛS3;Q) would be at least

two dimensional, which is not the case. If the minimal index, Index(N) = 2i, i > 1,

then H2(ΛS3;Q) = · · · = H2i−1(ΛS3;Q) = 0, which is not the case. That there is

at most one critical manifold of a given index is clear by an argument similar to the

one in the proof of Lemma 3.2.

24

Chapter 4

Orientability of Negative Bundles

We want to use the fact that E is perfect to conclude that all negative bundles over

the three dimensional critical manifolds are indeed orientable. Notice that this is

not a circular argument, since the proof of perfectness does not use orientability.

Proposition 4.1. There do not exist any one dimensional critical manifolds.

Proof. This is similar to the argument in the proof of Lemma 3.2. By S1-equivariant

perfectness of E there can be at most two circles of index 2j. Let 2k, k ≥ 2 be the

minimal index of a one dimensional critical manifold. This critical manifold con-

tributes a Q2 in degree 2k. We must then �ll the 2k− 3 gaps from H2S1(ΛS3, S3;Q)

to H2k−2S1 (ΛS3, S3;Q) with contributions that come in pairs or quartets, clearly im-

possible. If k = 1 we get a contradiction since H2S1(ΛS3, S3;Q) = Q

Corollary 4.1. The negative bundles over the three dimensional critical manifolds

are (S1-equivariantly) orientable.

25

Proof. First note that if the dimension of a critical manifold N is three, the action

of S1/Zn on N is free, since there are no one dimensional critical manifolds.

Hence the S1-Borel construction N×S1ES1 is homotopy equivalent to N/(S1/Zn)

and by the proof of Proposition 3.3 N/(S1/Zn) is homotopy equivalent to S2. Since

S2 is simply connected the bundle is orientable. We conclude that the negative

bundles are also oriented as ordinary vector bundles.

It was shown in the previous chapter that the negative bundles over the one and

�ve dimensional critical manifolds are orientable, so this corollary �nishes the proof

that all negative bundles are orientable.

26

Chapter 5

Topology of the Three Dimensional

Critical Manifolds

By calculating the Euler class of the bundle S1 → N → N/S1 = S2 we will deduce

that the three dimensional critical manifolds are integral cohomology three spheres

or lens spaces.

Theorem 2. Assume that N is a three dimensional critical manifold. The quotient

N/S1 = S2 is endowed with a symplectic structure which corresponds to the Euler

class of the S1-bundle S1 → N → N/S1; in particular the Euler class is nonzero

and N has the integral cohomology of either the three sphere or a lens space.

Proof. Note that by Corollary 3.1 N is connected. Parts of the proof rely on an

argument in [Bes78]; see [Bes78, De�nition 1.23, Proposition 2.11 and 2.16]. We

will describe some extra structure on the �xed point set N . In [Bes78, Chapter

27

2] the author considers a manifold all of whose geodesics are closed with the same

least period 2π. In that case the action of S1 on T 1S3 is free and one gets a

principal S1 bundle p : T 1S3 → T 1S3/S1. There is a canonical connection α ∈

H1dR

(T 1S3; L(S1)) on T 1S3 constructed as follows: Let pTS3 : TTS3 → TS3 be the

projection and TpS3 : TTS3 → TS3 be the tangent map. For X ∈ TTS3 we de�ne

α̃(X) = g(TpS3 (X), pTS3(X)). This form is the pullback to the tangent bundle of

the canonical one form on the cotangent bundle. De�ne a horizontal distribution on

TT 1S3 by Qu = {X ∈ TuT 1S3 | α(X) = 0}. Then TuT 1S3 = RZ ⊕ Qu, where Z

is the geodesic vector �eld on TT 1S3, and the corresponding connection form is α̃

restricted to T 1S3 which we denote by α. The Lie algebra of S1 is abelian and dα

is horizontal, so the curvature form of α is dα. Since the two form dα is invariant

under the action of S1, it is basic. The form dα is also the restriction to the unit

tangent bundle of the pullback to the tangent bundle of the canonical two form

on the cotangent bundle. Hence dα is nonzero on the complement of Z on every

TuT1S3, u ∈ T 1S3. By Chern-Weil Theory there exists ω ∈ H2

dR(T 1S3/S1) such

that p∗(ω) = dα. By [KN69, Theorem 5.1] this class is the Euler class of the bundle.

By [Bes78, Proposition 2.11] the class ω is a symplectic form on T 1S3/S1 (ω being

degenerate would mean that dα = 0 on a nonempty subset of the complement of Z,

which is not the case). Hence the Euler class is nonzero.

We now carry the argument over to the three dimensional critical manifolds.

By Proposition 4.1 we know that the action of S1 on N is free and that N/S1 is

28

homeomorphic to S2. Consider the principal bundle S1 → N → S2, with projection

q. We want to conclude that the Euler class of this bundle is nonzero. First note that

the geodesic vector �eld is tangent to the �xed point set. Consider the restriction of

α to TN and de�ne as above a distribution on TN by Q̃u = {X ∈ TuN | α(X) = 0}.

Then we have TN = RZ ⊕ Q̃u, since Z is a vector �eld on TN with α(Z) = 1; see

[Bes78, 1.57]. The restriction of α to TN is invariant under the action of S1 given by

the geodesic �ow, since by [Bes78, 1.56] LZα = 0, so the distribution Q̃ is invariant

under the action of S1. Since the map u 7→ Q̃u is clearly smooth, Q̃ is a horizontal

distribution and since α(Z) = 1, α is the connection of the distribution. Similarly

to the above, we see that since the Lie algebra of S1 is abelian the curvature of

the bundle is dα (dα is horizontal since dα(Z,−) = 0 by [Bes78, 1.56]). As dα is

horizontal and invariant under the action of S1 (LZdα = 0 by [Bes78, 1.57]), it is

basic. Hence we can �nd a form ω ∈ H2dR

(S2) such that q∗(ω) = dα. Thus, to see

that the quotient is symplectic, it su�ces to show that the curvature is nonzero on

Q̃. This follows from the following general statement about symplectic reduction.

The proof of this statement is that, similarly to the proof of Lemma 3.1, the +1

eigenspace of the Poincaré map is a symplectic subspace.

Lemma 5.1. Let (V 4, τ) be a symplectic vector space and let Zn act on V by linear

symplectic transformations. Assume that dim Fix(Zn) = 2. Then Fix(Zn) is a

symplectic subspace.

Consider the tangent space to T 1S3 at a point u ∈ T 1S3. This splits as a direct

29

sum Z⊕Qu and dα 6= 0 on Qu and on the four dimensional subspace the form dα is a

symplectic two form and the di�erential of the geodesic �ow acts by symplectic linear

transformations. The tangent space TuN also splits as the direct sum of Z ⊕ Q̃u,

Q̃u ⊂ TuN . It follows from the above lemma that the form dα restricted to TuN is

nonzero and hence, as above, that the form ω ∈ H2dR

(S2) makes the quotient into a

symplectic manifold. Again by [KN69, Theorem 5.1] the class ω is the Euler class

of the S1-bundle S1 → N → S2. Since the Euler class is nonzero an application of

the Gysin sequence for the bundle S1 → N → S2 shows that N is either an integral

cohomology three sphere or lens space.

Corollary 5.1. There are no three dimensional critical manifolds with the integral

cohomology of S3 or a lens space S3/Zr, r odd.

Proof. By Corollary 3.1 there exists at most one connected, critical manifold, N ,

of a given index. Let n be the multiplicity of a geodesic in N , i.e. c has length

2π/n. The action of O(2) leaves N invariant and hence induces an action on N

which is e�ectively free since there are no one dimensional critical manifolds. Let

Zn ⊆ S1 be the ine�ective kernel of the action. By identifying S1/Zn with S1 we

see that O(2)/Zn ∼= O(2). The group O(2)/Zn acts freely on N , so in particular

Z2 × Z2 ⊆ O(2)/Zn acts freely on N . By a a theorem of Smith [Bre72, Theorem

8.1] Z2-cohomology spheres do not support a free action of Z2×Z2, so since S3/Zr,

r odd, are Z2-cohomology spheres we have �nished the proof.

30

Chapter 6

Index Growth

In this section we will use the results of [BTZ82] concerning the Bott Iteration

Formula to obtain complete information about the index growth of geodesics in the

three and �ve dimensional critical manifolds. The Bott Iteration Formula states that

Index(cq) =∑

zq=1 I(θ), where I : S1 → R is a locally constant function, which will

be determined in the following. We will write I(θ) for I(z), z = eiθ and θ ∈ [0, 2π].

Notice that since the critical manifolds are nondegenerate in the sense of Bott,

a geodesic in a �ve dimensional critical manifold has Poincaré map equal to the

identity, and the Poincaré map of a geodesic in a three dimensional critical manifold

has a two by two identity block.

Since there exists a common period for the geodesics, we know that the Poincaré

map P of a closed geodesic is a root of unity. Let c be a closed geodesic of multiplicity

n. If there exists an eigenvalue e2πi/k where k|n, the map P k has two 2× 2 identity

31

blocks and since ck, k < n, still lies in a three dimensional critical manifold this

contradicts the fact that all three dimensional critical manifolds are nondegenerate.

Hence the eigenvalues of P are 1 and e2πi/n.

The splitting numbers S±(θ) are given by S±(θ) = limρ→0± I(θ + ρ) − I(θ), for

0 ≤ θ ≤ π, and completely determine the function I, since we know that I(θ) =

I(2π − θ), I is locally constant and that I only jumps at eigenvalues of P , i.e. at

a value of θ where S± is nonzero. If with respect to a symplectic basis the Jordan

normal form of the Poincaré map contains a 2× 2 block of the form cos(θ) σ sin(θ)−σ sin(θ) cos(θ)

for σ ∈ {±1} and 0 < θ < π, we say that the sign of the Jordan block is σ.

We now calculate the splitting numbers S+(0), S−(0), S+(2π/n) and S−(2π/n).

We note that in the case of a �ve dimensional critical manifold there is only one

eigenvalue 1 and the splitting numbers corresponding to that eigenvalue are 2.

Lemma 6.1. Let n be the multiplicity of a closed geodesic c in a three dimensional

critical manifold and let P be its Poincaré map. The splitting numbers of P , S+(0)

and S−(0) are both equal to 1. The splitting numbers S+(2π/n) and S−(2π/n)

depend on the sign σ of the Jordan block and for σ = +1 we have S+(2π/n) = 1,

S−(2π/n) = 0 and for σ = −1 we have S+(2π/n) = 0, S−(2π/n) = 1. If n = 2 the

splitting numbers are all equal to 1.

Proof. This follows directly from [BTZ82, Theorem 2.13] and the calculation of the

32

Jordan blocks, [BTZ82, page 222].

Next we want to determine the function I, which is quite simple knowing the

splitting numbers.

Proposition 6.1. Let c be a geodesic of index l in a three dimensional critical

manifold. The function I : S1 → R is for σ = 1 given by

I(θ) =

l, for θ = 0,

l + 1 for θ ∈ (0, 2π/n],

l + 2 for θ ∈ (2π/n, π].

If σ = −1 or n = 2, I is given by

I(θ) =

l, for θ = 0,

l + 1 for θ ∈ (0, 2π/n),

l for θ ∈ [2π/n, π].

Proof. This follows directly from the previous lemma.

Remark. The index of the iterate ck, k = 2, 3, . . ., of a geodesic c in a three dimen-

sional critical manifold of multiplicity n and some given index can now be calculated

from Proposition 6.1 and the Bott Iteration Formula Index(cq) =∑

zq=1 I(θ). In the

case of a �ve dimensional critical manifold, 1 is the only one eigenvalue and the

corresponding splitting numbers are 2.

33

Chapter 7

Reductions of the Berger Conjecture

in Dimension Three

Note that by Chapter 4 the negative bundles are oriented. The �rst consequence of

the preceding chapters is the following theorem.

Theorem 3. Let g be a metric on S3 all of whose geodesics are closed. The geodesics

have the same least period if and only if the energy function is perfect for ordinary

cohomology.

Proof. By [Zil77] the cohomology ring H∗(ΛS3;Z) contains no torsion, so if E is

perfect Corollary 5.1 implies that T 1S3 is the only critical manifold and hence that

all geodesics are closed of the same least period. If all geodesics are closed of the

same least period the only critical manifold is T 1S3. Theorem 1 implies that the

indices must be 2(2k − 1), k ≥ 1, which implies that E is perfect.

34

For reductions of the conjecture, we �rst consider the case where Index(T 1S3) =

2. If the action of S1 has an isotropy subgroup Zq there exists a three dimensional

critical manifold Fix(Zq) ⊂ T 1S3. Since by the Bott Iteration Formula we have

Index(cq) ≥ Index(c) we see that the three dimensional critical manifold must also

have index two, since the minimal index of a critical manifold consisting of noncon-

stant geodesics is two, by Corollary 3.1. This is a contradiction since there exists at

most one critical manifold of a given index, by Corollary 3.1.

Next let c be a geodesic of index two which lies in a three dimensional critical

manifold N and such that c has length 2π/n. We have three cases to consider: Either

n = 2 or n > 2 and the sign σ of the nontrivial Jordan block of N is ±1. By the

Bott Iteration Formula we have Index(c2) = I(π) + Index(c). Hence it follows from

Proposition 6.1 that if n = 2 or n > 2 and σ = −1 that Index(c2) = 4. Furthermore,

if σ = +1, Index(c2) = 6.

Now we assume that the sectional curvature K satis�es a/4 ≤ K ≤ a. It was

proved in [BTZ83] that

H∗(Λ16π2/2aS3, Λ4π

2/2aS3;Z) = H∗−2(T1S3;Z), (7.1)

which has a Z in degree 2, 4, 5 and 7. By the curvature assumption and since S3

is simply connected, we know that the injectivity radius is greater than or equal to

π/√

a. Hence there exist no closed geodesics of length less than 2π/√

a and hence

of energy less than 4π2/2a. Also note that c2 does not lie in Λ16π2/2aS3. Indeed,

if E(c) = 4π2/2a and hence E(c2) = 16π2/2a a theorem of Tsukamoto implies that

35

the metric g has constant sectional curvature; see [Tsu66]. This contradicts our

assumption that there exist geodesics that are shorter than the common period 2π.

Hence by Equation 7.1 the geodesics in N must lie in (Λ16π2/2aS3, Λ4π

2/2aS3) since

by Corollary 3.1 N is the only critical manifold of index two.

Next we consider the case where Index(c2) = 4. By Theorem 2 we know that N

contributes a rational class in degree two and �ve. Hence by Equation 7.1 there must

exist another simple closed geodesic of index four in Λ16π2/2aS3. Since Index(c2) = 4

we have two distinct critical manifolds of index four, contradicting Corollary 3.1.

We summarize in a theorem.

Theorem 4. Let g be a metric on S3 all of whose geodesics are closed with sec-

tional curvature K satisfying a/4 ≤ K ≤ a. Then all geodesics have the same least

period unless the unique closed geodesic c ∈ (Λ16π2/2aS3, Λ4π2/2aS3) of index two has

Index(c2) = 6, i.e. the Poincaré map of c contains a rotation with sign σ = +1

Remark. If Index(c2) = 6 we conclude from Equation 7.1 that there must exist a

geodesic d of index four, which must lie in a three dimensional critical manifold M ,

since otherwise H6S1(ΛS3, S3;Q) = Q3. Assume that the length of d is 2π/m. If

we iterate the geodesic c a multiple of n times we land in T 1S3. The index of ck,

k = 2, 3, . . . of c is given by Proposition 6.1. When calculating the index of dk,

k = 2, 3, . . ., of the geodesic d there are two cases to consider: σ = ±1. For some

combinations of n and m we are able to derive a contradiction by showing that two

distinct critical manifolds must contribute in the same degree.

36

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